Geodesic
Solution of time-dependent Schr\"odinger equation in a~nontraditional Hilbert space
Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 1, pp. 86-96

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A Hilbert space that is not traditional for quantum mechanics is used. It is the space of square-integrable functions of not only the space and spin variables $\mathbf r$ but also the time $t$ on an interval $(-T/2, T/2)$. In this space, the time-dependent Schrödinger equation formally reduces to a time-independent equation, which can be solved by the methods of time-independent scattering theory. A solution is found for cases in which a time-dependent perturbation contains several harmonics; this perturbation is small, and the duration of the perturbation is appreciably shorter than the lifetime of the excited levels. 
@article{TMF_1991_87_1_a8,
     author = {A. Ya. Dzyublik},
     title = {Solution of time-dependent {Schr\"odinger} equation in a~nontraditional {Hilbert} space},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {86--96},
     publisher = {mathdoc},
     volume = {87},
     number = {1},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1991_87_1_a8/}
}
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A. Ya. Dzyublik. Solution of time-dependent Schr\"odinger equation in a~nontraditional Hilbert space. Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 1, pp. 86-96. http://geodesic.mathdoc.fr/item/TMF_1991_87_1_a8/